Draws a 3d immersion of a Klein bottle that smoothly deforms between the Etruscan Venus surface, the Roman surface, the Boy surface surface, and the Ida surface.
etruscanvenus [--display host:display.screen] [--install] [--visual visual] [--window] [--root] [--window-id number] [--delay usecs] [--fps] [--mode display-mode] [--wireframe] [--surface] [--transparent] [--appearance appearance] [--solid] [--distance-bands] [--direction-bands] [--colors color-scheme] [--onesided-colors] [--twosided-colors] [--distance-colors] [--direction-colors] [--no-change-colors] [--view-mode view-mode] [--walk] [--turn] [--no-deform] [--deformation-speed float] [--initial-deformation float] [--etruscan-venus] [--roman] [--boy] [--ida] [--orientation-marks] [--projection mode] [--perspective] [--orthographic] [--speed-x float] [--speed-y float] [--speed-z float] [--walk-direction float] [--walk-speed float]
The etruscanvenus program shows a 3d immersion of a Klein bottle that smoothly deforms between the Etruscan Venus surface, the Roman surface, the Boy surface, and the Ida surface. You can walk on the Klein bottle or turn it in 3d. Topologically, all surfaces are Klein bottles, even the Roman and Boy surfaces, which are doubly covered and therefore appear to be an immersed real projective plane. The smooth deformation between these surfaces was constructed by George K. Francis.
The Klein bottle is a non-orientable surface. To make this apparent, the two-sided color mode can be used. Alternatively, orientation markers (curling arrows) can be drawn as a texture map on the surface of the Klein bottle. While walking on the Klein bottle, you will notice that the orientation of the curling arrows changes (which it must because the Klein bottle is non-orientable). Since all the surfaces except the Ida surface have points where the surface normal is not well defined for some points, walking is only performed on the Ida surface.
As mentioned above, the Roman and Boy surfaces are doubly covered and therefore appear to be an immersed real projective plane. Since some of the parameter names are based on this interpretation of the surface, the geometry of the real projective plane will be briefly disussed. The real projective plane is a model for the projective geometry in 2d space. One point can be singled out as the origin. A line can be singled out as the line at infinity, i.e., a line that lies at an infinite distance to the origin. The line at infinity is topologically a circle. Points on the line at infinity are also used to model directions in projective geometry. Direction and distance bands refer to this interpretation of the surface. If direction bands are used, the bands extend from the origin of the projective plane in different directions to the line at infinity and back to the origin. If distance bands are used, the bands lie at constant distances to the origin. The same interpretation is used for distance and direction colors. Although there is no conceptually equivalent geometric interpretation for the two Klein bottle surfaces (the Etruscan Venus and Ida surfaces), the smooth deformation between the surfaces results in a natural extension of these concepts to the Klein bottle surfaces.
The immersed surfaces can be projected to the screen either perspectively or orthographically. When using the walking mode, perspective projection to the screen will be used.
There are three display modes for the Klein bottle: mesh (wireframe), solid, or transparent. Furthermore, the appearance of the surface can be as a solid object or as a set of see-through bands. The bands can be distance bands or direction bands, as explained above.
The colors with with the surface is drawn can be set to one-sided, two-sided, distance, or direction. In one-sided mode, the surface is drawn with the same color on both sides of the underlying triangles. In two-sided mode, the surface is drawn with one color on one side of the underlying triangles and the complementary color on the other side. Since the surface actually only has one side, the color jumps from red to green along a line on the surface. This mode enables you to see that the surface is non-orientable. In distance mode, the surface is displayed with fully saturated colors that depend on the distance of the points on the projective plane to the origin, as described above. If the surface is displayed as distance bands, each band will be displayed with a different color. In direction mode, the surface is displayed with fully saturated colors that depend on the angle of the points on the projective plane with respect to the origin (see above for an explanation). If the surface is displayed as direction bands, each band will be displayed with a different color. The colors used to color the surface can either be static or can be changed dynamically.
The rotation speed for each of the three coordinate axes around which the Klein bottle rotates can be chosen.
Furthermore, in the walking mode the walking direction in the 2d base square of the surface and the walking speed can be chosen. The walking direction is measured as an angle in degrees in the 2d square that forms the coordinate system of the surface. A value of 0 or 180 means that the walk is along a circle at a randomly chosen distance from the origin (parallel to a distance band). A value of 90 or 270 means that the walk is directly along a direction band. Any other value results in a curved path along the surface. As noted above, walking is performed only on the Ida surface.
By default, the immersion of the Klein bottle smoothly deforms between the Etruscan Venus surface, the Roman surface, the Boy surface, and the Ida surface. It is possible to choose the speed of the deformation. Furthermore, it is possible to switch the deformation off. It is also possible to determine the initial deformation of the immersion. This is mostly useful if the deformation is switched off, in which case it will determine the appearance of the surface. A value of 0 corresponds to the Etruscan Venus surface, a value of 1000 to the Roman surface, a value of 2000 to the Boy surface, and a value of 3000 to the Ida surface.
This program is inspired by George K. Francis's book "A Topological Picturebook", Springer, 1987, by George K. Francis's paper "The Etruscan Venus" in P. Concus, R. Finn, and D. A. Hoffman: "Geometric Analysis and Computer Graphics", Springer, 1991, and by a video entitled "The Etruscan Venus" by Donna J. Cox, George K. Francis, and Raymond L. Idaszak, presented at SIGGRAPH 1989.
etruscanvenus accepts the following options:
The following four options are mutually exclusive. They determine how the Klein bottle is displayed.
The following four options are mutually exclusive. They determine the appearance of the Klein bottle.
The following five options are mutually exclusive. They determine how to color the Klein bottle.
The following options determine whether the colors with which the Klein bottle are displayed are static or are changing dynamically.
The following three options are mutually exclusive. They determine how to view the Klein bottle.
The following options determine whether the surface is being deformed.
The following option determines the deformation speed.
The following options determine the initial deformation of the surface. As described above, this is mostly useful if --no-deform is specified.
The following options determine whether orientation marks are shown on the Klein bottle.
The following three options are mutually exclusive. They determine how the Klain bottle is projected from 3d to 2d (i.e., to the screen).
The following three options determine the rotation speed of the Klein bottle around the three possible axes. The rotation speed is measured in degrees per frame. The speeds should be set to relatively small values, e.g., less than 4 in magnitude. In walk mode, all speeds are ignored.
The following two options determine the walking speed and direction.
If you run this program in standalone mode in its turn mode, you can rotate the Klein bottle by dragging the mouse while pressing the left mouse button. This rotates the Klein bottle in 3d. To examine the Klein bottle at your leisure, it is best to set all speeds to 0. Otherwise, the Klein bottle will rotate while the left mouse button is not pressed. This kind of interaction is not available in the walk mode.
Copyright © 2019-2020 by Carsten Steger. Permission to use, copy, modify, distribute, and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. No representations are made about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty.
Carsten Steger <carsten@mirsanmir.org>, 05-jan-2020.
